Chapter 21: SPECT and the Physics of Collimation

In the previous chapter we put PET and SPECT side by side and saw how PET fixes direction through coincidence detection. This chapter narrows in on SPECT. Reconstructing the activity distribution that SPECT measures rides directly on the tools we built for CT (FBP, or the OSEM of Chapter 8). The reason SPECT still earns its own chapter is that its image quality is decided almost entirely by a single sheet of lead: the collimator. Here the basic trade-off of imaging-system design (resolution and sensitivity cannot both be had) appears in its simplest form.

Why a collimator is needed

PET counted the pair of 511 keV photons flying in exactly opposite directions and fixed the direction from the line (the LOR) joining the two detection points. The tracer a SPECT study uses (such as $^{99m}$Tc) emits a single gamma ray. With no partner photon, one detection alone cannot tell us which direction it came from.

So SPECT places a lead collimator in front of the detector: a plate perforated by countless fine parallel holes that passes only the gamma rays aligned with the holes and absorbs the obliquely incident ones in the lead walls (the septa). Direction is selected not by electronic coincidence but by mechanical absorption. This approach (throwing away most of the gamma rays to obtain direction) is the main reason SPECT's sensitivity is more than an order of magnitude below PET's.

Resolution worsens with distance

The geometric resolution of a parallel-hole collimator is set by how far the source is from the plate. With hole diameter $d$, effective hole length $L_\text{eff}$, and source-to-collimator distance $z$, the blur of a point source (full width at half maximum) is approximately

$$R_\text{col} \;=\; \frac{d\,(L_\text{eff} + z)}{L_\text{eff}}$$

At $z=0$ (flush against the plate) $R_\text{col}=d$, the hole diameter itself. The farther the source, the more the image blurs by the angular spread a single hole accepts, and $R_\text{col}$ grows linearly with $z$. Unlike the focal blur of CT this is not magnification but plain degradation, which is why SPECT keeps the detector as close to the body as possible while it rotates.

The detector itself also has an intrinsic resolution $R_\text{int}$ (from the crystal and position logic), and the resolution of the whole system combines the two in quadrature:

$$R_\text{sys} \;=\; \sqrt{R_\text{col}^2 + R_\text{int}^2}$$

At short range $R_\text{int}$ still matters, but a little farther out $R_\text{col}$ dominates. This is why SPECT image quality is said to be decided by the collimator.

Raising resolution lowers sensitivity

So why not just make the holes thinner and longer to shrink $R_\text{col}$ without limit? Because the thinner and longer the hole, the narrower the solid angle of gamma rays it accepts, and the fewer photons it detects (that is, sensitivity, or geometric efficiency, falls). The efficiency is approximately

$$g \;\approx\; \left(\frac{K\,d}{L_\text{eff}}\right)^{\!2}\!\left(\frac{d}{d+t}\right)^{\!2}$$

where $t$ is the septal thickness and $K$ a hole-shape constant (about 0.26 for hexagonal holes). Efficiency scales as $(d/L_\text{eff})^2$, so doubling $L_\text{eff}$ to improve resolution drops sensitivity to a quarter. Efficiency is almost independent of distance $z$ (farther away fewer rays reach a given hole, but more holes come into view and the two cancel). Only resolution degrades with distance; sensitivity stays put. That is the character of parallel-hole geometry.

Lower sensitivity means fewer counts collected in the same time. Radioactive counting obeys Poisson statistics, so fewer counts make a grainier image. A high-resolution collimator is therefore "sharp but noisy," a high-sensitivity one "smooth but blurred." Clinically, low-energy high-resolution (LEHR) and high-sensitivity collimators are chosen to match the task.

Simulation: moving sharpness and noise together

A phantom with hot spots and a pair of closely spaced line sources is blurred by the system resolution and given Poisson noise whose count level is proportional to sensitivity. Lengthening the hole $L_\text{eff}$ shrinks $R_\text{sys}$ until the two line sources separate, but the relative sensitivity drops and the image grows grainy. Increasing the source distance $z$ lets you confirm the other half: resolution alone degrades while sensitivity holds. The plot on the right shows how the current collimator's resolution worsens linearly with distance.

The reconstruction itself is the same as CT

Once the collimator has selected direction, projections at each angle (a sinogram) are in hand and reconstruction is no different from CT. In practice, even for this modality where attenuation and scatter matter, the mainstay is the OSEM of Chapter 8 with attenuation and resolution correction folded in. What makes SPECT special is the entrance to the measurement (how a single photon is given a direction), and once past that, the skeleton of the inverse problem is shared. That is the view this textbook has kept returning to.

References

• Cherry SR, Sorenson JA, Phelps ME. Physics in Nuclear Medicine, 4th ed. Elsevier (2012) — standard derivation of collimator resolution and sensitivity.
• Anger HO. Scintillation Camera. Review of Scientific Instruments 29, 27–33 (1958) — the original gamma camera.
• Gunter DL. Collimator characteristics and design. In Nuclear Medicine, Henkin RE et al. (eds.), Mosby (1996).
• Bruyant PP. Analytic and Iterative Reconstruction Algorithms in SPECT. Journal of Nuclear Medicine 43, 1343–1358 (2002).